| import Mathlib |
| import OAI.Probability.Ballisticity.Estimates.IntegerMedian |
|
|
| namespace OAI |
|
|
| section |
| section |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology |
| namespace DirectionalTransience |
|
|
| def signedHeight {d : ℕ} (e : Direction d) (x : Lattice d) : ℤ := |
| if e.2 then x e.1 else -x e.1 |
|
|
| lemma signedHeight_projection {d : ℕ} (e : Direction d) (x : Lattice d) : |
| dot (realPosition x) (realPosition (step e)) = (signedHeight e x : ℝ) := by |
| rw [dot_signed_direction] |
| simp only [signedCoordinate,signedHeight,Int.cast_ite,Int.cast_neg] |
|
|
| lemma signedHeight_step_le {d : ℕ} (e : Direction d) (x : Lattice d) (f : Direction d) : |
| signedHeight e (x+step f) ≤ signedHeight e x+1 := by |
| simp only [signedHeight,Pi.add_apply,step] |
| split <;> split <;> (try split) <;> omega |
|
|
| theorem transverse_common_gaussian_marginal {d : ℕ} (ν : Measure (Row d)) |
| [IsProbabilityMeasure ν] (hue : UniformElliptic ν) (e f : Direction d) (hef : e.1 ≠ f.1) |
| (htrans : DirectionallyTransient ν (realPosition (step e))) |
| (r : ℕ → ℝ) (hr : IsGaussianSequence (independentConditionedPairLaw ν (realPosition (step e))) |
| (commonIncrementProcess (realPosition (step e)) f 0) r) (τ : ℝ≥0) : |
| letI : IsProbabilityMeasure (independentConditionedPairLaw ν (realPosition (step e))) := |
| independentConditionedPairLaw_probability ν _ |
| (ne_of_gt (noDrop_positive_of_directionallyTransient ν _ htrans)) |
| TendstoInDistribution |
| (fun i P => realPartialSum (fun k => commonIncrementProcess (realPosition (step e)) f k P) |
| (⌊(τ:ℝ)*fluctuationScale (independentConditionedPairLaw ν (realPosition (step e))) |
| (commonIncrementProcess (realPosition (step e)) f 0) (r i)⌋₊) / r i) |
| atTop id (fun _ => independentConditionedPairLaw ν (realPosition (step e))) (gaussianReal 0 τ) := |
| independent_common_gaussian_marginal ν hue _ (signed_direction_unit e) htrans |
| (signedHeight e) (signedHeight_projection e) (signedHeight_step_le e) f |
| (independent_commonWordIncrement_nonzero ν hue e f hef htrans) r hr τ |
|
|
| lemma max_fourth_increment {m a : ℝ} : |
| (max m |a|)^4-m^4 ≤ 2*|a| *((max m |a|)^3-m^3) := by |
| by_cases h : |a| ≤ m |
| · rw [max_eq_left h]; ring_nf; rfl |
| · rw [max_eq_right (le_of_not_ge h)] |
| have hp := mul_nonneg (mul_nonneg (abs_nonneg a) (sq_nonneg m)) |
| (sub_nonneg.mpr (le_of_not_ge h)) |
| nlinarith [sq_nonneg (|a|^2-m^2)] |
|
|
| lemma partialSumMax_fourth_pathwise (X : ℕ → ℝ) (n : ℕ) : |
| (partialSumMax X n)^4 ≤ 2*(partialSumMax X n)^3*|realPartialSum X n|- |
| 2*∑ k ∈ Finset.range n, (partialSumMax X k)^3*absSlope (realPartialSum X k)*X k := by |
| induction n with |
| | zero => simp [partialSumMax] |
| | succ n ih => |
| have hincr := max_fourth_increment (m := partialSumMax X n) |
| (a := realPartialSum X (n+1)) |
| have hs := mul_le_mul_of_nonneg_left (abs_supporting (realPartialSum X n) (X n)) |
| (pow_nonneg (partialSumMax_nonneg X n) 3) |
| rw [← realPartialSum_succ] at hs |
| rw [Finset.sum_range_succ] |
| change (max (partialSumMax X n) |realPartialSum X (n+1)|)^4 ≤ _ |
| change _ ≤ 2*(max (partialSumMax X n) |realPartialSum X (n+1)|)^3 * |realPartialSum X (n+1)|-_ |
| nlinarith |
|
|
| lemma partialSumMax_fourth_pathwise_bound (X : ℕ → ℝ) (n : ℕ) : |
| (partialSumMax X n)^4 ≤ 32*(realPartialSum X n)^4- |
| 4*∑ k ∈ Finset.range n, (partialSumMax X k)^3*absSlope (realPartialSum X k)*X k := by |
| have h := partialSumMax_fourth_pathwise X n |
| have hy : 2*(partialSumMax X n)^3*|realPartialSum X n| ≤ |
| (partialSumMax X n)^4/2+16*(realPartialSum X n)^4 := by |
| nlinarith [sq_nonneg ((partialSumMax X n)^2-4*partialSumMax X n*|realPartialSum X n|), |
| sq_nonneg ((partialSumMax X n)^2-8*(realPartialSum X n)^2), |
| sq_abs (realPartialSum X n)] |
| linarith |
|
|
| lemma integrable_partialSumMax_pow {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsFiniteMeasure μ] (X : ℕ → Ω → ℝ) (hX : ∀ k, Measurable (X k)) |
| {z : ℝ} (hz : 0 ≤ z) (hbound : ∀ k ω, |X k ω| ≤ z) (n p : ℕ) : |
| Integrable (fun ω => (partialSumMax (fun k => X k ω) n)^p) μ := by |
| apply integrable_of_abs_bound μ _ ((measurable_partialSumMax X hX n).pow_const p) ((n*z)^p) |
| intro ω |
| rw [abs_of_nonneg (pow_nonneg (partialSumMax_nonneg _ _) _)] |
| exact pow_le_pow_left₀ (partialSumMax_nonneg _ _) (partialSumMax_bound _ hz (fun k => hbound k ω) n) p |
|
|
| lemma integrable_realPartialSum_pow {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsFiniteMeasure μ] (X : ℕ → Ω → ℝ) (hX : ∀ k, Measurable (X k)) |
| {z : ℝ} (hz : 0 ≤ z) (hbound : ∀ k ω, |X k ω| ≤ z) (n p : ℕ) : |
| Integrable (fun ω => (realPartialSum (fun k => X k ω) n)^p) μ := by |
| apply integrable_of_abs_bound μ _ ((measurable_realPartialSum X hX n).pow_const p) ((n*z)^p) |
| intro ω |
| rw [abs_pow] |
| exact pow_le_pow_left₀ (abs_nonneg _) ((abs_partialSum_le_max _ _).trans |
| (partialSumMax_bound _ hz (fun k => hbound k ω) n)) p |
|
|
| lemma absSlope_cube (t : ℝ) : (absSlope t)^3 = absSlope t := by |
| unfold absSlope |
| split <;> norm_num |
|
|
| lemma independent_max_cube_transform {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) |
| (X : ℕ → Ω → ℝ) (hX : ∀ k, Measurable (X k)) (hind : iIndepFun X μ) (n : ℕ) : |
| IndepFun (fun ω => (partialSumMax (fun k => X k ω) n)^3*absSlope (realPartialSum (fun k => X k ω) n)) |
| (X n) μ := by |
| have hh := (independent_max_transform μ X hX hind n).comp |
| (show Measurable (fun x : ℝ => x^3) by fun_prop) measurable_id |
| simpa only [Function.comp_def,mul_pow,absSlope_cube,id_eq] using hh |
|
|
| lemma integral_partialSumMax_fourth_le {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (X : ℕ → Ω → ℝ) (hX : ∀ k, Measurable (X k)) |
| (hind : iIndepFun X μ) (hmean : ∀ k, ∫ ω, X k ω ∂μ = 0) |
| {z : ℝ} (hz : 0 ≤ z) (hbound : ∀ k ω, |X k ω| ≤ z) (n : ℕ) : |
| (∫ ω, (partialSumMax (fun k => X k ω) n)^4 ∂μ) ≤ |
| 32*∫ ω, (realPartialSum (fun k => X k ω) n)^4 ∂μ := by |
| let C (k : ℕ) (ω : Ω) := (partialSumMax (fun j => X j ω) k)^3*absSlope (realPartialSum (fun j => X j ω) k) |
| have hC (k : ℕ) : Measurable (C k) := ((measurable_partialSumMax X hX k).pow_const 3).mul |
| (measurable_absSlope.comp (measurable_realPartialSum X hX k)) |
| have hCI (k : ℕ) : Integrable (fun ω => C k ω*X k ω) μ := by |
| apply integrable_of_abs_bound μ _ ((hC k).mul (hX k)) ((k*z)^3*z) |
| intro ω |
| dsimp [C] |
| rw [abs_mul,abs_mul,abs_absSlope,mul_one,abs_of_nonneg (pow_nonneg (partialSumMax_nonneg _ _) _)] |
| exact mul_le_mul (pow_le_pow_left₀ (partialSumMax_nonneg _ _) |
| (partialSumMax_bound _ hz (fun j => hbound j ω) k) 3) (hbound k ω) |
| (abs_nonneg _) (pow_nonneg (mul_nonneg (Nat.cast_nonneg _) hz) 3) |
| have hzero (k : ℕ) : ∫ ω, C k ω*X k ω ∂μ = 0 := by |
| rw [(independent_max_cube_transform μ X hX hind k).integral_fun_mul_eq_mul_integral |
| (hC k).aestronglyMeasurable (hX k).aestronglyMeasurable,hmean,mul_zero] |
| have hm := integral_mono (integrable_partialSumMax_pow μ X hX hz hbound n 4) |
| (((integrable_realPartialSum_pow μ X hX hz hbound n 4).const_mul 32).sub |
| ((integrable_finsetSum (Finset.range n) (fun k _ => hCI k)).const_mul 4)) |
| (fun ω => partialSumMax_fourth_pathwise_bound (fun k => X k ω) n) |
| simp only [Pi.sub_apply] at hm |
| rw [integral_sub ((integrable_realPartialSum_pow μ X hX hz hbound n 4).const_mul 32) |
| ((integrable_finsetSum (Finset.range n) (fun k _ => hCI k)).const_mul 4), |
| integral_const_mul,integral_const_mul,integral_finsetSum _ (fun k _ => hCI k)] at hm |
| simpa only [hzero,Finset.sum_const_zero,mul_zero,sub_zero] using hm |
|
|
| lemma independent_partialSum_current {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) |
| (X : ℕ → Ω → ℝ) (hX : ∀ k, Measurable (X k)) (hind : iIndepFun X μ) (n : ℕ) : |
| IndepFun (fun ω => realPartialSum (fun k => X k ω) n) (X n) μ := by |
| have hh := (independent_past_current μ X hX hind n).comp |
| (show Measurable (fun v : Fin n → ℝ => ∑ k, v k) by fun_prop) measurable_id |
| convert hh using 1 |
| · funext ω |
| exact (Fin.sum_univ_eq_sum_range (fun k => X k ω) n).symm |
| · rfl |
|
|
| lemma integrable_pow_mul_of_abs_bound {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsFiniteMeasure μ] (F G : Ω → ℝ) |
| (hF : Measurable F) (hG : Measurable G) {a b : ℝ} |
| (ha : ∀ ω, |F ω| ≤ a) (hb : ∀ ω, |G ω| ≤ b) (p q : ℕ) : |
| Integrable (fun ω => F ω^p*G ω^q) μ := by |
| apply integrable_of_abs_bound μ _ (by fun_prop) (a^p*b^q) |
| intro ω |
| rw [abs_mul,abs_pow,abs_pow] |
| exact mul_le_mul (pow_le_pow_left₀ (abs_nonneg _) (ha ω) p) |
| (pow_le_pow_left₀ (abs_nonneg _) (hb ω) q) (pow_nonneg (abs_nonneg _) _) |
| (pow_nonneg ((abs_nonneg (F ω)).trans (ha ω)) _) |
|
|
| lemma integral_independent_add_fourth {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (F G : Ω → ℝ) |
| (hF : Measurable F) (hG : Measurable G) (hind : IndepFun F G μ) |
| (hmF : ∫ ω, F ω ∂μ = 0) (hmG : ∫ ω, G ω ∂μ = 0) |
| {a b : ℝ} (ha : ∀ ω, |F ω| ≤ a) (hb : ∀ ω, |G ω| ≤ b) : |
| (∫ ω, (F ω+G ω)^4 ∂μ) = |
| (∫ ω, F ω^4 ∂μ)+6*(∫ ω, F ω^2 ∂μ)*(∫ ω, G ω^2 ∂μ)+(∫ ω, G ω^4 ∂μ) := by |
| have hI := integrable_pow_mul_of_abs_bound μ F G hF hG ha hb |
| have hF4 : Integrable (fun ω => F ω^4) μ := by simpa using hI 4 0 |
| have hG4 : Integrable (fun ω => G ω^4) μ := by simpa using hI 0 4 |
| have h31 : Integrable (fun ω => 4*(F ω^3*G ω^1)) μ := (hI 3 1).const_mul 4 |
| have h22 : Integrable (fun ω => 6*(F ω^2*G ω^2)) μ := (hI 2 2).const_mul 6 |
| have h13 : Integrable (fun ω => 4*(F ω^1*G ω^3)) μ := (hI 1 3).const_mul 4 |
| have hfactor (p q : ℕ) : (∫ ω, F ω^p*G ω^q ∂μ) = |
| (∫ ω, F ω^p ∂μ)*(∫ ω, G ω^q ∂μ) := |
| (hind.comp (by fun_prop : Measurable (fun x : ℝ => x^p)) |
| (by fun_prop : Measurable (fun x : ℝ => x^q))).integral_fun_mul_eq_mul_integral |
| (hF.pow_const p).aestronglyMeasurable (hG.pow_const q).aestronglyMeasurable |
| have heq : (fun ω => (F ω+G ω)^4) = |
| (fun ω => F ω^4+4*(F ω^3*G ω^1)+6*(F ω^2*G ω^2)+4*(F ω^1*G ω^3)+G ω^4) := by |
| ext ω; ring |
| have h12 : Integrable (fun ω => F ω^4+4*(F ω^3*G ω^1)) μ := hF4.add h31 |
| have h123 : Integrable (fun ω => F ω^4+4*(F ω^3*G ω^1)+6*(F ω^2*G ω^2)) μ := h12.add h22 |
| have h1234 : Integrable (fun ω => F ω^4+4*(F ω^3*G ω^1)+6*(F ω^2*G ω^2)+4*(F ω^1*G ω^3)) μ := h123.add h13 |
| rw [heq,integral_add h1234 hG4,integral_add h123 h13,integral_add h12 h22, |
| integral_add hF4 h31] |
| simp only [integral_const_mul] |
| rw [hfactor 3 1,hfactor 2 2,hfactor 1 3] |
| simp only [pow_one,hmF,hmG,mul_zero,zero_mul,add_zero] |
| ring |
|
|
| lemma integral_iid_partialSum_fourth_le {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (X : ℕ → Ω → ℝ) (hX : ∀ k, Measurable (X k)) |
| (hind : iIndepFun X μ) (hident : ∀ k, IdentDistrib (X k) (X 0) μ μ) |
| (hmean : ∫ ω, X 0 ω ∂μ = 0) |
| {z : ℝ} (hz : 0 ≤ z) (hbound : ∀ k ω, |X k ω| ≤ z) (n : ℕ) : |
| (∫ ω, (realPartialSum (fun k => X k ω) n)^4 ∂μ) ≤ |
| n*(∫ ω, (X 0 ω)^4 ∂μ)+3*(n:ℝ)^2*(∫ ω, (X 0 ω)^2 ∂μ)^2 := by |
| have hmeans (k : ℕ) : ∫ ω, X k ω ∂μ = 0 := (hident k).integral_eq.trans hmean |
| have hpows (k p : ℕ) : (∫ ω, (X k ω)^p ∂μ) = (∫ ω, (X 0 ω)^p ∂μ) := |
| ((hident k).comp (u := fun x : ℝ => x^p) (by fun_prop)).integral_eq |
| have hsq (n : ℕ) : (∫ ω, (realPartialSum (fun k => X k ω) n)^2 ∂μ) = |
| n*(∫ ω, (X 0 ω)^2 ∂μ) := by |
| rw [integral_partialSum_square_eq_sum μ X hX hind hmeans hbound] |
| simp only [hpows,Finset.sum_const,Finset.card_range,nsmul_eq_mul] |
| have hsummean (n : ℕ) : ∫ ω, realPartialSum (fun k => X k ω) n ∂μ = 0 := by |
| rw [show (fun ω => realPartialSum (fun k => X k ω) n) = |
| (fun ω => ∑ k ∈ Finset.range n, X k ω) from rfl, |
| integral_finsetSum _ (fun k _ => integrable_of_abs_bound μ _ (hX k) z (hbound k))] |
| simp only [hmeans,Finset.sum_const_zero] |
| induction n with |
| | zero => simp [realPartialSum] |
| | succ n ih => |
| simp only [realPartialSum_succ] |
| rw [integral_independent_add_fourth μ _ _ (measurable_realPartialSum X hX n) (hX n) |
| (independent_partialSum_current μ X hX hind n) (hsummean n) (hmeans n) |
| (fun ω => (abs_partialSum_le_max _ _).trans (partialSumMax_bound _ hz (fun k => hbound k ω) n)) |
| (hbound n),hsq] |
| simp only [hpows] |
| push_cast |
| nlinarith [sq_nonneg (∫ ω, (X 0 ω)^2 ∂μ)] |
|
|
| lemma integral_iid_truncated_max_fourth_le {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (X : ℕ → Ω → ℝ) (hX : ∀ k, Measurable (X k)) |
| (hind : iIndepFun X μ) (hident : ∀ k, IdentDistrib (X k) (X 0) μ μ) |
| (hsym : IdentDistrib (X 0) (fun ω => -X 0 ω) μ μ) |
| {z r : ℝ} (hz : 0 ≤ z) (hzr : z ≤ r) (n : ℕ) : |
| (∫ ω, (partialSumMax (fun k => symmetricTruncate z (X k ω)) n)^4 ∂μ) ≤ |
| 32*(n:ℝ)*z^2*truncatedVariance μ (X 0) r+ |
| 96*(n:ℝ)^2*(truncatedVariance μ (X 0) r)^2 := by |
| let Y := fun k ω => symmetricTruncate z (X k ω) |
| have hY (k : ℕ) : Measurable (Y k) := (measurable_symmetricTruncate z).comp (hX k) |
| have hYb := fun k ω => symmetricTruncate_bound hz (X k ω) |
| have hYi : iIndepFun Y μ := hind.comp (fun _ => symmetricTruncate z) (fun _ => measurable_symmetricTruncate z) |
| have hYid (k : ℕ) : IdentDistrib (Y k) (Y 0) μ μ := (hident k).comp (measurable_symmetricTruncate z) |
| have hYm : ∫ ω, Y 0 ω ∂μ = 0 := integral_symmetricTruncate_zero μ (X 0) z hsym |
| have hYmeans (k : ℕ) : ∫ ω, Y k ω ∂μ = 0 := (hYid k).integral_eq.trans hYm |
| have hI (p : ℕ) : Integrable (fun ω => (Y 0 ω)^p) μ := by |
| apply integrable_of_abs_bound μ _ ((hY 0).pow_const p) (z^p) |
| intro ω |
| rw [abs_pow] |
| exact pow_le_pow_left₀ (abs_nonneg _) (hYb 0 ω) p |
| have hvar : (∫ ω, (Y 0 ω)^2 ∂μ) ≤ truncatedVariance μ (X 0) r := by |
| apply integral_mono (hI 2) (integrable_truncated_square μ (X 0) (hX 0) r) |
| intro ω |
| exact (symmetricTruncate_sq_le z _).trans |
| (min_le_min le_rfl (pow_le_pow_left₀ hz hzr 2)) |
| have hfour : (∫ ω, (Y 0 ω)^4 ∂μ) ≤ z^2*(∫ ω, (Y 0 ω)^2 ∂μ) := by |
| rw [← integral_const_mul] |
| apply integral_mono (hI 4) ((hI 2).const_mul (z^2)) |
| intro ω |
| have hs : (Y 0 ω)^2 ≤ z^2 := by |
| nlinarith [sq_abs (Y 0 ω),hYb 0 ω,abs_nonneg (Y 0 ω)] |
| nlinarith [mul_le_mul_of_nonneg_right hs (sq_nonneg (Y 0 ω))] |
| have hv0 : 0 ≤ ∫ ω, (Y 0 ω)^2 ∂μ := integral_nonneg fun _ => sq_nonneg _ |
| calc |
| _ ≤ 32*(∫ ω, (realPartialSum (fun k => Y k ω) n)^4 ∂μ) := |
| integral_partialSumMax_fourth_le μ Y hY hYi hYmeans hz hYb n |
| _ ≤ 32*(n*(∫ ω, (Y 0 ω)^4 ∂μ)+3*(n:ℝ)^2*(∫ ω, (Y 0 ω)^2 ∂μ)^2) := |
| mul_le_mul_of_nonneg_left (integral_iid_partialSum_fourth_le μ Y hY hYi hYid hYm hz hYb n) (by norm_num) |
| _ ≤ 32*(n*(z^2*truncatedVariance μ (X 0) r)+3*(n:ℝ)^2*(truncatedVariance μ (X 0) r)^2) := by |
| gcongr |
| · exact hfour.trans (mul_le_mul_of_nonneg_left hvar (sq_nonneg z)) |
| _ = _ := by ring |
|
|
| lemma measureReal_truncated_max_gt_le {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (X : ℕ → Ω → ℝ) (hX : ∀ k, Measurable (X k)) |
| (hind : iIndepFun X μ) (hident : ∀ k, IdentDistrib (X k) (X 0) μ μ) |
| (hsym : IdentDistrib (X 0) (fun ω => -X 0 ω) μ μ) |
| {z r a : ℝ} (hz : 0 ≤ z) (hzr : z ≤ r) (ha : 0 < a) (n : ℕ) : |
| μ.real {ω | a < partialSumMax (fun k => symmetricTruncate z (X k ω)) n} ≤ |
| (32*(n:ℝ)*z^2*truncatedVariance μ (X 0) r+ |
| 96*(n:ℝ)^2*(truncatedVariance μ (X 0) r)^2) / a^4 := by |
| let Y := fun k ω => symmetricTruncate z (X k ω) |
| have hY (k : ℕ) : Measurable (Y k) := (measurable_symmetricTruncate z).comp (hX k) |
| have hYb := fun k ω => symmetricTruncate_bound hz (X k ω) |
| have hb := mul_meas_ge_le_integral_of_nonneg |
| (Eventually.of_forall (fun ω => pow_nonneg (partialSumMax_nonneg (fun k => Y k ω) n) 4)) |
| (integrable_partialSumMax_pow μ Y hY hz hYb n 4) (a^4) |
| have hmono : μ.real {ω | a < partialSumMax (fun k => Y k ω) n} ≤ |
| μ.real {ω | a^4 ≤ (partialSumMax (fun k => Y k ω) n)^4} := by |
| apply measureReal_mono _ (measure_ne_top _ _) |
| intro ω hω |
| exact pow_le_pow_left₀ ha.le hω.le 4 |
| apply (le_div_iff₀ (pow_pos ha 4)).mpr |
| calc |
| _ ≤ a^4*μ.real {ω | a^4 ≤ (partialSumMax (fun k => Y k ω) n)^4} := by |
| simpa only [mul_comm] using mul_le_mul_of_nonneg_left hmono (pow_nonneg ha.le 4) |
| _ ≤ (∫ ω, (partialSumMax (fun k => Y k ω) n)^4 ∂μ) := hb |
| _ ≤ _ := integral_iid_truncated_max_fourth_le μ X hX hind hident hsym hz hzr n |
|
|
| lemma measureReal_iid_max_gt_fourth_bound {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (X : ℕ → Ω → ℝ) (hX : ∀ k, Measurable (X k)) |
| (hind : iIndepFun X μ) (hident : ∀ k, IdentDistrib (X k) (X 0) μ μ) |
| (hsym : IdentDistrib (X 0) (fun ω => -X 0 ω) μ μ) |
| {z r a : ℝ} (hz : 0 ≤ z) (hzr : z ≤ r) (ha : 0 < a) (n : ℕ) : |
| μ.real {ω | a < partialSumMax (fun k => X k ω) n} ≤ |
| n*μ.real {ω | z < |X 0 ω|}+ |
| (32*(n:ℝ)*z^2*truncatedVariance μ (X 0) r+ |
| 96*(n:ℝ)^2*(truncatedVariance μ (X 0) r)^2) / a^4 := by |
| let A : ℕ → Set Ω := fun k => {ω | z < |X k ω|} |
| let B : Set Ω := {ω | a < partialSumMax (fun k => symmetricTruncate z (X k ω)) n} |
| have hsub : {ω | a < partialSumMax (fun k => X k ω) n} ⊆ |
| (⋃ k ∈ Finset.range n, A k) ∪ B := by |
| intro ω hω |
| by_cases ht : ω ∈ ⋃ k ∈ Finset.range n, A k |
| · exact Or.inl ht |
| · right |
| have hc : ∀ k < n, X k ω = symmetricTruncate z (X k ω) := by |
| intro k hk |
| have hnot : ¬ z < |X k ω| := fun h => ht (Set.mem_iUnion.mpr |
| ⟨k,Set.mem_iUnion.mpr ⟨Finset.mem_range.mpr hk,h⟩⟩) |
| simp only [symmetricTruncate,ite_eq_left (le_of_not_gt hnot)] |
| change a < partialSumMax (fun k => symmetricTruncate z (X k ω)) n |
| rwa [← partialSumMax_congr n hc] |
| have htail (k : ℕ) : μ.real (A k) = μ.real (A 0) := by |
| exact congrArg ENNReal.toReal ((hident k).measure_mem_eq |
| (show MeasurableSet {x : ℝ | z < |x|} from measurableSet_lt measurable_const measurable_id.abs)) |
| calc |
| _ ≤ μ.real ((⋃ k ∈ Finset.range n, A k) ∪ B) := measureReal_mono hsub (measure_ne_top _ _) |
| _ ≤ μ.real (⋃ k ∈ Finset.range n, A k)+μ.real B := measureReal_union_le _ _ |
| _ ≤ (∑ k ∈ Finset.range n, μ.real (A k))+μ.real B := |
| add_le_add (measureReal_biUnion_finset_le _ _) le_rfl |
| _ = n*μ.real {ω | z < |X 0 ω|}+μ.real B := by |
| simp only [htail,Finset.sum_const,Finset.card_range,nsmul_eq_mul,A] |
| _ ≤ _ := add_le_add le_rfl (measureReal_truncated_max_gt_le μ X hX hind hident hsym hz hzr ha n) |
|
|
| lemma gaussian_charFun_power_limit_count {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (S : Ω → ℝ) (hS : Measurable S) |
| (hI : Integrable S μ) (hne : 0 < μ {x | S x ≠ 0}) |
| (hsym : IdentDistrib S (fun x => -S x) μ μ) (r : ℕ → ℝ) |
| (hr : IsGaussianSequence μ S r) (k : ℕ → ℕ) {τ : ℝ} |
| (hk : Tendsto (fun i => (k i : ℝ)/fluctuationScale μ S (r i)) atTop (𝓝 τ)) (t : ℝ) : |
| Tendsto (fun i => (charFun (μ.map S) (t/r i))^(k i)) |
| atTop (𝓝 (Complex.exp (-(τ*t^2/2 : ℝ)))) := by |
| have hn : Tendsto (fun i => fluctuationScale μ S (r i)) atTop atTop := |
| (fluctuationScale_tendsto μ S hS hI hne).comp hr.1 |
| have hd := gaussian_cosine_drift μ S hS hne r hr t |
| have hzero : Tendsto (fun i => (∫ x, Real.cos (t*(S x/r i)) ∂μ)-1) atTop (𝓝 0) := by |
| have hz := hd.mul (tendsto_inv_atTop_zero.comp hn) |
| simp only [mul_zero] at hz |
| apply hz.congr' |
| filter_upwards [hn.eventually (eventually_gt_atTop (0:ℝ))] with i hi |
| dsimp only [Function.comp_def] |
| field_simp |
| have ha : Tendsto (fun i => ∫ x, Real.cos (t*(S x/r i)) ∂μ) atTop (𝓝 1) := by |
| simpa using hzero.add_const 1 |
| have hdk : Tendsto (fun i => (k i : ℝ)* |
| ((∫ x, Real.cos (t*(S x/r i)) ∂μ)-1)) atTop (𝓝 (-(τ*t^2/2))) := by |
| have hc := hk.mul hd |
| have he : τ * (-(t^2/2)) = -(τ*t^2/2) := by ring |
| rw [he] at hc |
| apply hc.congr' |
| filter_upwards [hn.eventually (eventually_gt_atTop (0:ℝ))] with i hi |
| field_simp |
| convert (tendsto_pow_variable_of_drift ha hdk).ofReal using 1 |
| · ext i |
| rw [symmetric_charFun_cosine μ S hS hsym, ← Complex.ofReal_pow] |
| congr 2 |
| apply integral_congr_ae |
| filter_upwards [] with x |
| congr 1 |
| ring |
| · simp |
|
|
| lemma charFun_independent_finite_blocks {Ω ι : Type*} [MeasurableSpace Ω] [Fintype ι] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (X : ℕ → Ω → ℝ) |
| (hX : ∀ k, Measurable (X k)) (hind : iIndepFun X μ) |
| (hident : ∀ k, IdentDistrib (X k) (X 0) μ μ) |
| (k : ι → ℕ) (g : (j : ι) → Fin (k j) → ℕ) |
| (hg : Function.Injective (fun p : (j : ι) × Fin (k j) => g p.1 p.2)) (r : ℝ) |
| (t : EuclideanSpace ℝ ι) : |
| charFun (μ.map (fun ω => WithLp.toLp 2 (fun j => (∑ l, X (g j l) ω)/r))) t = |
| ∏ j, (charFun (μ.map (X 0)) (t j/r))^(k j) := by |
| classical |
| let Y (p : (j : ι) × Fin (k j)) (ω : Ω) := (t p.1/r)*X (g p.1 p.2) ω |
| have hY p : Measurable (Y p) := (hX _).const_mul _ |
| have hiY : iIndepFun Y μ := (hind.precomp hg).comp |
| (fun p x => (t p.1/r)*x) (fun _ => by fun_prop) |
| have heq : charFun (μ.map (fun ω => WithLp.toLp 2 (fun j => (∑ l, X (g j l) ω)/r))) t = |
| charFun (μ.map (fun ω => ∑ p, Y p ω)) 1 := by |
| rw [charFun, charFun, integral_map (by fun_prop) (by fun_prop), |
| integral_map (by fun_prop) (by fun_prop)] |
| apply integral_congr_ae |
| filter_upwards [] with ω |
| congr 2 |
| simp only [PiLp.inner_apply, RCLike.inner_apply, conj_trivial, |
| one_mul, Fintype.sum_sigma, Y] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro j _ |
| rw [← Finset.mul_sum] |
| ring |
| rw [heq,hiY.charFun_map_fun_sum_eq_prod (fun p => (hY p).aemeasurable)] |
| simp only [Finset.prod_apply] |
| rw [Fintype.prod_sigma] |
| apply Finset.prod_congr rfl |
| intro j _ |
| have hterm (l : Fin (k j)) : charFun (μ.map (Y ⟨j,l⟩)) 1 = |
| charFun (μ.map (X 0)) (t j/r) := by |
| dsimp [Y] |
| rw [charFun_map_mul_comp (hX _).aemeasurable,mul_one,(hident _).map_eq] |
| simp only [hterm,Finset.prod_const,Finset.card_univ,Fintype.card_fin] |
|
|
| theorem gaussian_iid_finite_block_limit {Ω ι : Type*} [MeasurableSpace Ω] [Fintype ι] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (X : ℕ → Ω → ℝ) |
| (hX : ∀ k, Measurable (X k)) (hind : iIndepFun X μ) |
| (hident : ∀ k, IdentDistrib (X k) (X 0) μ μ) |
| (hI : Integrable (X 0) μ) (hne : 0 < μ {x | X 0 x ≠ 0}) |
| (hsym : IdentDistrib (X 0) (fun x => -X 0 x) μ μ) |
| (r : ℕ → ℝ) (hr : IsGaussianSequence μ (X 0) r) |
| (k : ℕ → ι → ℕ) (g : (i : ℕ) → (j : ι) → Fin (k i j) → ℕ) |
| (hg : ∀ i, Function.Injective (fun p : (j : ι) × Fin (k i j) => g i p.1 p.2)) |
| (τ : ι → ℝ≥0) |
| (hk : ∀ j, Tendsto (fun i => (k i j : ℝ)/fluctuationScale μ (X 0) (r i)) atTop (𝓝 (τ j : ℝ))) : |
| TendstoInDistribution |
| (fun i ω => WithLp.toLp 2 (fun j => (∑ l, X (g i j l) ω)/r i)) atTop |
| (WithLp.toLp 2 : (ι → ℝ) → EuclideanSpace ℝ ι) (fun _ => μ) |
| (Measure.pi (fun j => gaussianReal 0 (τ j))) where |
| forall_aemeasurable _ := by fun_prop |
| aemeasurable_limit := by fun_prop |
| tendsto := by |
| refine ProbabilityMeasure.tendsto_iff_tendsto_charFun.2 fun t => ?_ |
| change Tendsto (fun i => charFun (μ.map (fun ω => WithLp.toLp 2 (fun j => (∑ l, X (g i j l) ω)/r i))) t) |
| atTop (𝓝 (charFun ((Measure.pi (fun j => gaussianReal 0 (τ j))).map (WithLp.toLp 2)) t)) |
| simp only [charFun_independent_finite_blocks μ X hX hind hident _ _ (hg _) _,charFun_pi] |
| apply tendsto_finsetProd _ |
| intro j _ |
| simpa only [charFun_gaussianReal,Complex.ofReal_zero,mul_zero,zero_mul,zero_sub, |
| Complex.ofReal_neg,Complex.ofReal_div,Complex.ofReal_mul,Complex.ofReal_pow, |
| Complex.ofReal_ofNat] using |
| gaussian_charFun_power_limit_count μ (X 0) (hX 0) hI hne hsym r hr (fun i => k i j) (hk j) (t j) |
|
|
| end DirectionalTransience |
| end |
| end |
|
|
| end OAI |
|
|
